by
Lial, Margaret L.; Hornsby, John; McGinnis, Terry

Answer

$\sqrt{m}-2$

Work Step by Step

$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\dfrac{m-4}{\sqrt{m}+2}
,$ multiply the numerator and the denominator by the conjugate of the denominator. Then use special products to simplify the result.
$\bf{\text{Solution Details:}}$
Multiplying the numerator and the denominator of the given expression by the conjugate of the denominator, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{m-4}{\sqrt{m}+2} \cdot\dfrac{\sqrt{m}-2}{\sqrt{m}-2}
\\\\=
\dfrac{(m-4)(\sqrt{m}-2)}{(\sqrt{m}+2)(\sqrt{m}-2)}
.\end{array}
Using the product of the sum and difference of like terms which is given by $(a+b)(a-b)=a^2-b^2,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\dfrac{(m-4)(\sqrt{m}-2)}{(\sqrt{m})^2-(2)^2}
\\\\=
\dfrac{(m-4)(\sqrt{m}-2)}{m-4}
\\\\=
\dfrac{(\cancel{m-4})(\sqrt{m}-2)}{\cancel{m-4}}
\\\\=
\sqrt{m}-2
.\end{array}